Abstract

We discuss the use of some new time transfer techniques for computing TAI time links. Precise point positioning (PPP) uses GPS dual frequency carrier phase and code measurements to compute the link between a local clock and a reference time scale with the precision of the carrier phase and the accuracy of the code. The time link between any two stations can then be computed by a simple difference. We show that this technique is well adapted and has better short-term stability than other techniques used in TAI. We present a method of combining PPP and two-way time transfer that takes advantage of the qualities of each technique, and shows that it would bring significant improvement to TAI links.

1. Introduction

GPS carrier phase measurements are two orders of magnitude more precise than the
GPS code data, much less sensitive to propagation multipaths, and allow a
better estimate of the atmosphere effects. Receivers able to measure phase and
code are becoming commonplace in time laboratories. For this reason, the CCTF,
at its 17th meeting in September 2006 [1] passed a recommendation “Concerning
the use of Global Navigation Satellite System (GNSS) carrier phase techniques
for time and frequency transfer in International Atomic Time (TAI),” in which
it asked that “the International Bureau of Weights and Measures (BIPM), in a
highly cooperative manner, generate its own solutions, make them freely
available to others, and add them to its time transfer comparison database,”
and that “the BIPM begin preparing software and techniques for introduction of
the data into the computation of Circular T” (excerpts from Recommendation CCTF 4, 2006).

One GPS carrier phase analysis technique is precise point positioning (PPP), in
which dual frequency phase and code measurement are used to compare the
reference clock of a single receiver to a reference time scale. This is possible
thanks to the precise satellite orbits and clocks provided by the International
GNSS Service (IGS) [2], where the reference time scale is IGS time [3]. In
recent years, the time community has shown a growing interest in this technique
and a number of time transfer experiments have been carried out (see, e.g., [4–6]).

PPP allows one to compute UTC(k)-IGS Time, where UTC(k) is the local reference
of any laboratory participating in TAI, which is equipped with such a
geodetic-type receiver. Then any link [UTC(k)-UTC(l)] can be
computed by simple difference. This approach makes sense for computing TAI time
links because it can be applied for any individual laboratory, properly
equipped, without the need to participate in an organized network such as the
IGS. Although participation in the IGS has other practical advantages such as
improved experience return and reliability, it is unlikely that all
laboratories in the TAI network will participate, so that the PPP approach is
particularly adequate. In addition, it is easy to be put into operation using one of the several
existing software packages. PPP is the natural follower of the all-in-view
technique [7], which has recently been introduced to compute TAI links, in
which code-only measurements are used to compute UTC(k)-GPS Time. PPP combines the precise GPS phase
and the accurate code measurements, thus can provide excellent short-term
stability from the phase and good accuracy.

In Section 2, we recall the basis of the PPP technique and
in Section 3, we present the practical implementation of PPP used at the BIPM.
In Section 4, we present a combination method that allow a merging of the
independent techniques PPP and two-way time transfer (TW). In Section 5, we
present results of time link
comparisons between PPP, TW, and the PPP+TW combination that allow an estimation of the stability of each technique and a determination of
an optimal processing of TAI time links.

2. Basic Principles of Precise Point Positioning

The observation equations for the carrier phases 𝐿𝑗
at each the two GPS frequencies (𝑓𝑗, j = 1,2) and the pseudoranges
(𝑃𝑗), here both expressed in meters, can be written as
𝐿𝑗𝑐=𝑇𝑝+𝜏𝑡−𝜏𝑖,𝑗−𝜏𝑠+𝜏𝑟+𝑁𝑗𝜆𝑗𝑐+𝜏𝜙,𝑗+𝜀𝜙,𝑗,𝑃𝑗𝑐=𝑇𝑝+𝜏𝑡+𝜏𝑖,𝑗−𝜏𝑠+𝜏𝑟+𝜏𝐶,𝑗+𝜀𝐶,𝑗,(1)
where c is the velocity of light, 𝑇𝑝 is the coordinate time of propagation of the signal from the satellite to the
receiver in empty space (between the instantaneous positions of the antenna
phase centers), and 𝜏𝑡 and 𝜏𝑖 are the
propagation delays due to the troposphere and the ionosphere,
respectively. 𝜏𝑠 is the satellite clock error, 𝜏𝑟 is
the receiver clock error, 𝜆𝑗 is the carrier wavelength, 𝑁𝑗 is the phase ambiguity, 𝜏𝐶 is the instrumental code delay, 𝜏𝜙
is the instrumental phase shift on the
carrier, and 𝜀𝜙 and 𝜀𝐶 represent the phase and code errors, respectively.
In the PPP processing, the
measurements (1) are processed without differencing and it is necessary
to use precise satellite position (from the IGS) and precise satellite clock
value with respect to a given time reference (IGS time). In this case,
𝑇𝑝 is a function of the receiver’s position 𝑋𝑟 only,
and 𝜏𝑠 disappears from (1), where 𝜏𝑟(𝑡) is now referenced to IGS time. Note that errors in the IGS
precise orbits and clocks will then contribute to the measurement errors in (2)
and (3) below. In addition, the ionosphere-free linear combinations, 𝐿3=(𝑓21𝐿1−𝑓22𝐿2)/(𝑓21−𝑓22), 𝑃3=(𝑓21𝑃1−𝑓22𝑃2)/, are used, so that (𝑓21−𝑓22) disappears. Note that
ionospheric effects of higher orders, which are not removed by the linear
combination, also end up in the measurement errors. The unknown code delay
corresponding to the linear combination, 𝜏𝑖,
is included in the receiver clock term 𝜏𝐶,3 if
we consider that PPP is used only
to provide a stable link. However, calibration can provide an estimate of 𝜏𝑟(𝑡)
(see some discussion on the achievable accuracy of PPP in Section 4). The tropospheric delay 𝜏𝐶 (t, direction) is expressed as 𝜏𝑡,
where 𝜏𝑧(𝑡)𝑀𝑓 is the delay at zenith and 𝜏𝑧 is a given
mapping function. Note that the actual tropospheric model distinguishes a hydrostatic
and a wet component, but this distinction is not necessary in this short
presentation. After the ionosphere-free linear combination, the phase
ambiguities together with the unknown carrier phase shift are considered as a
real-valued phase ambiguity for each satellite pass, here noted 𝑀𝑓.
Equation (1) is then rewritten as
𝑋3
which show explicitly the parameters which have to be determined by PPP, that is, the
receiver position 𝐿3𝑐=𝑇𝑝𝑋𝑟+𝜏𝑧𝑡𝑀𝑓+𝜏𝑟𝑡+𝑋3+𝜀𝜙,3,𝑃(2)3𝑐=𝑇𝑝𝑋𝑟+𝜏𝑧𝑡𝑀𝑓+𝜏𝑟𝑡+𝜀𝐶,3,(3), the receiver clock 𝑋𝑟, the zenith tropospheric delay 𝜏𝑟(𝑡),
and the real-valued ambiguities. These parameters are obtained in a global adjustment, where a priori uncertainties are assigned
to the observations, of order, a few nanoseconds for the codes (3) and, of
order, tens of picoseconds for the phase observations (2). For a more complete
presentation of PPP (see the geodetic literature, e.g., [8, 9]).

3. Operation of PPP for TAI Link Computation

The software presently in operation at the BIPM is the GPSPPP software developed by
Natural Resources Canada [8]. Until early 2007 data processing (such as
those presented in Section 5) were performed with the version 1365, released in
June 2005, while more recent treatments use version 1087, and released in May
2007. GPSPPP follows the lines presented in Section 2, and also
includes some specific features adapted to the time transfer such as the
possibility to allow for a clock process noise in solving 𝜏𝑧(𝑡) and the continuous processing of an “unlimited” (in principle) number of days
in a single run. In addition, the recent (1087) release adopts the new IGS
paradigm of absolute antenna phase center offsets [10, 11], in usage since November
2006, and uses updated models for station displacements (e.g., ocean tidal
loading) and troposphere mapping function.

The feature which allows long continuous runs is
particularly interesting for TAI, where computations are done at the beginning of each month for the
whole preceding month, so that one month is the natural processing interval. In
this case, it is not necessary to go to smaller computation intervals, like in
the quasi-real-time IGS processing, which necessitates short (daily) batches. A
well-known feature of GPS phase and code solutions based on successive (e.g.,
daily) batches is the presence of the so-called “boundary discontinuities”
[12]. It is to be noted that we do not expect our “long batch” solution to
remove boundary discontinuities, as if they were somehow an artifact of an
analysis procedure. The discontinuities originate in the noise of the GPS code
measurements, and long batches would decrease them if they are due to pure
white noise processes. However, it has been shown that other noise processes are
present and sometimes dominant [13], so that discontinuities do not necessarily
decrease, and may even increase, as the batch duration increases. Nevertheless, long batches are
preferable to daily batches because, in the latter case, the daily discontinuities
(of order 100 picoseconds or more) greatly affect the stability estimates at
averaging times of interest to clock analysis (hours to days). As long-term
systematic effects in code measurements may still affect results of our monthly
batches, we expect to eventually see them by comparison to other independent
time transfer techniques.

The main operational parameters of the GPSPPP software for the computation of 1-month
batches for TAI links are the following: we use IGS final SP3 orbits and
5-minute SV clocks (note that, for a real TAI computation carried out at the beginning of each
month, the latency of the final IGS products (14 days) is too long so that the
IGS rapid products must be used); a
priori data weights are taken as 1 m for pseudorange, and 1 cm for phase; the elevation cutoff is set to 10°; the observation sampling and clock solution interval are
both 5 minutes; the tropospheric zenith delay is modeled as 3 mm/𝜏𝑟(𝑡) random
walk, with the Niell mapping function [14] used; ocean loading coefficients are
from the Chalmers Centre for Astrophysics and Space Science (http://www.oso.chalmers.se/~loading);
station coordinates are estimated on each 1-month batch.

With the recent 1087 release, additional operational
parameters include the use of the IGS file igs05.atx for satellite antenna
offset values and station antenna phase center variations [10, 11], and the use of
the VMF1 mapping function [15], without horizontal gradient estimated and
updated models for ocean loading coefficients.

4. Combination of PPP with Two-Way Time Transfer

PPP or any technique including code and carrier phase (CP) provides time transfer results
with high sampling rates due to the automatic acquisition and analysis
procedures, and very good short-term stability, due to the low noise of the CP
measurements. However, the long-term stability and accuracy depend on the code
measurements which are subject to variations in the multipath environment, in
the hardware of the receiving equipment, or to residual propagation effects
(although these should be at a much lower level). It is estimated that the
accuracy of a GPS dual-frequency link is limited to a few nanoseconds by the
calibration techniques used [16], but the long-term stability may be at a lower
level. On the other hand, TW techniques generally have relatively low sampling
rates due to the cost and complexity of the observation procedures. It may have
a better long-term stability than GPS because of the low level of multipaths
and of the cancellation of several effects in the two-way operation but no
clear evidence is yet available. However, it seems that the calibration of TW
links can be performed with higher accuracy than for GPS links; for TW
calibration, the uncertainty is at or below 1 nanosecond [17], although the
calibration exercise is difficult and expensive. Considering that TW presently
provides the most accurate link, we expect to develop a combination technique
that would allow obtaining the short-term stability of PPP or any CP-based link
with the accuracy of the TW link.

The combination of TW and a CP-based technique can be considered as a typical
estimation problem, where a function y(t)
must be determined from two estimates Y and √hr: 𝑌 is
an estimate of y, provided by TW; and 𝑌 is an estimate
of the time derivative 𝑌, provided by the CP-based technique. Note that PPP also
provides an estimate of y due to the use of the codes, but we
will not make use of this fact in the following.

There are several existing methods to estimate a function using an estimate of its
derivative. One of the successful methods is that of Vondrak-Cepek [18], which
is based on the Whittaker-Robinson smoothing that removes the high-frequency
noise present in a series of unequally spaced observations. The goal is to find
a compromise between the smoothness of the searched function, on the one hand,
and the fidelity of this function to the observed values, on the other hand.
The original Vondrak smoothing [19, 20] considered only the fidelity to an estimate
of the function itself. It has largely been used in astrometry and in TAI
production. The Vondrak-Cepek smoothing also takes into account the fidelity to
an estimate of the function’s derivative by minimizing the quantity 𝑦,
where S measures the smoothness of the
searched function y, F its fidelity to the observed function Y,
and 𝑄=𝑆+𝜀𝐹+𝜀0𝐹0 the fidelity of 𝐹0 to the observed
function 𝑦. The coefficient 𝑌 quantifies the fidelity F, and 𝜀≥0 quantifies
the fidelity 𝜀0≥0. The coefficients can be considered as defining the
response of a filter with transfer functions 𝐹0 and 𝑇(𝑓),
respectively, where 𝑇0(𝑓)
and 𝑇(𝑓)=1/(1+(1/𝜀)(2𝜋𝑓)6). The dimensions of 𝑇0(𝑓)=1/(1+(1/𝜀0)(2𝜋𝑓)4) and 𝜀 are
time−6 and time−4, respectively. The transfer functions 𝜀0
and 𝑇(𝑓) represent the ratio between the amplitude after smoothing to the
amplitude before smoothing for a periodic function of frequency f.

5. Tests on TAI Time Links

We present time transfer results obtained on the link between the USNO (Washington DC, USA)
and the PTB (Braunschweig, Germany). This link is particularly suited for
comparisons because three independent techniques are operated on a regular
basis in these two laboratories. One (TW-Ku) is the typical Intelsat-based two-way link regularly used
for TAI, one (TW-X) is a special link operated by the USNO in the X-band, and
one (PPP) is from the PPP analysis of the data from geodetic GPS receivers. In
addition, time transfer based on single-frequency GPS receivers is also
available, but is not considered here because of the higher noise level due to
the single-frequency operation.

Results of several technique comparisons are presented below for this particular link.
In such a case (several independent techniques available), it is possible to
determine which of the techniques is more advantageous; however, our main goal
here is to test analysis strategies that may be applied to a number of TAI
links, particularly those where two techniques (TW-Ku and geodetic GPS) are
available. Indeed, considering that some 10 laboratories accounting for some
60% of the total clock weight in TAI are already in this case, it is of utmost
importance to devise a technique that provides the best stability and
robustness to TAI computation.

It is the analyst’s choice to determine the
transfer function coefficients best adapted to the problem at hand. In our
case, the goal is to give predominance to the CP-based technique for short-term
and to remove most of the possible diurnal disturbances in TW signals (see Section
5). This leads to choosing 𝑇0(𝑓) = 26400 d−6 and 𝜀 = 154000 d−4 (see [21] for details).

5.1. Estimation of the Stability of PPP and of Two Independent TW Techniques by Three-Corner Hat Analysis

Four consecutive months from the link USNO-PTB for the
three independent techniques (PPP, TW-Ku, and TW-X) have been analyzed and
compared. The data cover October 2005 to January 2006 and are nearly complete
except for three gaps, each a few days long, one in TW-Ku end October 2005, and
two in TW-X in January 2006. The TW-Ku data are generally available every 2
hours and the total number of data points is 1230 (average interval 2.4 hours),
while the TW-X are generally available every hour with a total number of points
of 2360 (average interval 1.2 hours). The PPP results are computed every 5
minutes and are interpolated at the exact dates of the TW observations for
comparisons. Direct comparisons of the three pairs of techniques yield the root-mean
square (RMS) of the results over the whole period, as
follows:𝜀0

As the comparison covers four months, the RMS values
are probably mostly driven by the long-term stability of the techniques
compared. It seems therefore that, over this period, TW-Ku would be the less
stable over the long term. As an example, the TW-X to TW-Ku comparison is shown
in Figure 1. In order to obtain more rigorous estimates of the stability of each of the
techniques, we apply the three-corner hat analysis [22] to the three pairs of
comparisons. The results are shown in Figure 2 for the modified Allan deviation
and in Figure 3 for the time deviation. We see that PPP appears as the most
stable technique at all averaging times until 3-4 days, with TW-X a close
second and TW-Ku a bit more unstable (note that the data points for this
comparison have been chosen to be those of TW-X, so that the TW-Ku data had to
be interpolated from a larger data interval; this must explain the bending of
the TW-Ku curve for the left-most point). Above 10-day averaging time, no valid
result can be obtained from this study, as the three-corner hat analysis
provides negative variances for most of the techniques and averaging times.

Figure 1: Comparison of TW-X and TW-Ku for the link USNO-PTB over 4 months. Top plot shows
the two results, middle plot shows the difference, and bottom plots show the modified Allan deviation
(left) and the time deviation (right) of the difference.

From the modified Allan deviation results, we estimate
that PPP and TW-X provide RMSUSNO-PTBTW-Ku−USNO-PTBPPP=1.2ns,RMSUSNO-PTBTW-X−USNO-PTBPPP=0.7ns,RMSUSNO-PTBTW-Ku−USNO-PTBTW-X=0.9ns.(4) uncertainty in frequency
comparison for averaging time at 3-4 days, while TW-Ku reaches this
performance for averaging time close to 10 days. The time deviation results are
directly derived from those in Figure 2, but reveal more clearly the level of
diurnal signatures (of order 100 picoseconds for PPP, 200 picoseconds for TW-X,
and more than 400 picoseconds for TW-Ku). They also show that PPP and TW-X
provide time stability below 200 picoseconds at all averaging times.

We conclude that PPP has very good short-term stability
and is, at least, as stable as TW-X (with 1 point every hour) for averaging time
up to a few days. PPP is typically more stable than TW-Ku (with 1 point every
two hours) for averaging time up to 10 days. The long-term stability (averaging
time 1×10−15 days) is more difficult to estimate, but the results presented
here indicate long-term instability typically similar to those of the TW
techniques. Nevertheless, systematic variations at a level of 1-2 nanoseconds over
durations of months are visible in PPP-TW comparisons, although their origin
cannot be readily determined.

5.2. Comparisons of PPP, TW, and PPP-TW Combination

Recent studies have already considered such combinations of TW with a GPS CP-based
technique [21, 23]. It has been shown that it may alleviate problems due to
diurnal disturbances in the TW-Ku band links (which is the major TW observation
in TAI). Figure 4 is a typical example where such diurnal signals, of 1–2.5 nanoseconds
peak to peak, are visible in the TW-Ku link. The time deviation (Figure 5(c))
obviously confirms the 1-day periodic term. Comparisons with PPP and TW-X
(Figures 5(a) and 5(b)) indicate that the diurnal signal originates indeed in the TW-Ku
technique. By combining the TW-Ku link with the PPP result, which is not
subject to such disturbance, we expect that the combined result will benefit
both from the short-term stability of the second technique and the accuracy of
the first technique. It can be shown (Figure 5(d)) that the combination indeed
provides the expected short-term stability. Comparing Figure 6 with Figure 4,
one can see that the combination has removed some fluctuations and one gap
associated with TW-Ku while preserving the link accuracy from TW-Ku.

Figure 4: Link USNO-PTB computed with TW-Ku over December 2005. Top and bottom right plots: link values in ns. Bottom left:
time deviation (unit is 0.1 ns).

Figure 5: Time deviation versus averaging time for the link USNO-PTB for various techniques: TW-X
(a); PPP (b); TW-Ku (c); combination PPP+TW-Ku (d).
Unit is 0.1 nanosecond. Note that TW-Ku has a time interval of 2 hours hence the first point is
for that averaging time.

Figure 6: Link USNO-PTB computed with the combination PPP+TW-Ku over December 2005.
Unit is 1 nanosecond.

Comparisons
between TW-X, on one hand, TW-Ku, PPP, and the combination PPP+TW-Ku, on the
other hand, have been carried out for several months starting October 2005. In all cases, it can
be shown that the combination provides performances which are equivalent or
better than each of the techniques. As an example, Table 1 summarizes 4 months
of comparisons by showing the RMS of the results of each comparison
over each month, and over the whole period. It can be seen that if we consider
TW-X to be the reference, the combination PPP+TW-Ku has equivalent or slightly
better performance than PPP only, and provides about 30% improvement with
respect to TW-Ku only. As already mentioned, it is worth performing the
combination in order to obtain the accuracy of the TW link and the robustness, for
example, with respect to gaps in either of the two techniques.

Table 1: Comparison of several time transfer techniques for the link USNO-PTB over 4 months. Values of the RMS of the
comparison results in nanoseconds.

6. Conclusions

The
computation of PPP links for TAI is under implementation at the BIPM. In this
paper, we have described the procedures for computation and the validation of
the results in comparison to the TW technique. As we have shown, the quality of
PPP time links is such that TAI would already benefit from its introduction
(PPP would then replace the presently used code-only P3 links). The comparison
with TW-Ku links is not as straightforward because PPP has a clear advantage at
short term, but may be subject to more instability in the long term and the
calibration uncertainty should be lower for TW links. We have shown that one
solution is to combine PPP with TW-Ku in order to
obtain the short- and medium-term stability of PPP and the accuracy of TW, and
we have described such a combination method that is readily usable. This would
make better usage of the high redundancy of the TAI worldwide network without
significantly complicating the computation procedures.

Acknowledgments

The authors would like to thank the Natural Resources Canada for providing the
GPSPPP software, particularly F. Lahaye for continuous support and maintenance of the
software. The authors acknowledge the data contributed by time laboratories
participating in TAI. This work is based partly on two papers presented
by the authors at the joint IEEE-FCS'07/EFTF'07 conference.

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